基于核重构思想的最小二乘配点型无网格方法

介绍重构核点法的基本原理和近似函数的构造方法，并基于核重构思想，应用配点法和最小二乘原理，离散微分方程，建立求解的代数方程，提出了一种基于核重构思想的最小二乘配点型无网格方法．与一般配点法相比，该方法的系数矩阵是有对称正定的，计算精度高，稳定性好．该方法的实施不需要背景网格，不需要进行高斯积分，与Galerkin法相比，具有计算量小、边界条件处理简单的特点，是一种真正的无网格法．对该方法构造过程中的近似函数及其导数的计算、修正函数的计算及方法的实现等问题进行了探讨．文中结合若干典型算例，检验了该方法的有效性．

Least-squre point collocation method based on reproducing kernal ideas

Meshless or meshfree method for numerical analysis is one prospective numerical method that was rapidly developed in last decade. It can avoid the dependency to the mesh of traditional numerical methods such as finite element method. It can eliminate, at least partly, the mesh and perform numerical analysis without any initial mesh generation and mesh regeneration. So it posses obvious advantages in dealing with problems which contain mesh irregularity or moving boundary conditions. It is one of the hottest research areas in the solid mechanics over the world. Meshless method can be mainly divided into, according to its discretizing principle, Galerkin-type and point collocated-type etc. The implementation of Galerkin-based meshless method needs a background mesh and Gauss integrals. So it is not a truly meshless method. While the point collocated meshless method is truly meshless method because its implementation does not need any background mesh and Gauss integrals. In this paper, the principle of reproducing kernel particle method and the constructing method of approximated functions was first reviewed. Then a new least-square point collocation method was presented based on the reproducing kernel ideas by the principle of least-square method and collocation method. The system coefficient matrix generated by this method has symmetric property compared to usual collocation methods and a stable and accurate numerical result can be obtained. This point collocation method is a true meshless method because no background mesh and no numerical integrals were required in its implementation. So it poses less computational cost and the imposition of boundary conditions is very straightforward and convenient compared to Galerkin-based one. The calculation of the approximation function and its derivatives, that of the correction function and its implementation were investigated in this paper. The validity was examined by several numerical examples.